Re: Problem with LaplaceTransform and InverseLaplaceTransform

*To*: mathgroup at smc.vnet.net*Subject*: [mg67541] Re: Problem with LaplaceTransform and InverseLaplaceTransform*From*: ab_def at prontomail.com*Date*: Fri, 30 Jun 2006 04:14:26 -0400 (EDT)*References*: <e7td5h$3kh$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

aXi wrote: > Can someone point me to site that would help me in using Mathematica > 5.2for purpose of calculations related to Systhems theory. I'm having > trouble with several topics... for example doing > InverseLaplaceTransform of function: > > (1/s) * (TanH[pi*s/2]) > > Thanks in advance! If you want the inverse Laplace transform of Tanh[Pi*s/2]/s, it can be found as follows. First take the transform of Tanh[Pi*s/2]: In[1]:= fs = Tanh[Pi*s/2]/s; In[2]:= InverseLaplaceTransform[s*fs, s, t] Out[2]= DiracDelta[t] + 2*Sum[(-1)^K$40*DiracDelta[(-K$40)*Pi + t], {K$40, 1, Infinity}] Then integrate the result: In[3]:= Integrate[DiracDelta[t], t] + 2*Sum[Integrate[(-1)^k*DiracDelta[t - Pi*k], t], {k, Infinity}] Out[3]= UnitStep[t] + (-1 + (-1)^Floor[t/Pi])*UnitStep[-1 + t/Pi] Strictly speaking, we need to evaluate Integrate[f[tau], {tau, 0-, t}], 'including' the zero in the integration range. It is easy to see that the result is equal to (-1)^Floor[t/Pi] or Sign[Sin[t]]. Here's a check: In[4]:= LaplaceTransform[Sign[Sin[t]], t, s] - fs // Simplify Out[4]= 0 Maxim Rytin m.r at inbox.ru